Dispersion of sound in air, with constant temperature and pressure, is very slight, increasing for very short wavelengths, and for very loud noises. Why? Because the rapid sequence of weak compression/decompression steps as the sound propagates are adiabatic, or energy-conserving, for the normal ranges of sound. This leaves the local pressure, temperature and density unchanged.
As a result, the equation for the speed of sound in an ideal gas is $c^2 = \gamma P/\rho$, with $c$ the speed of sound, $\gamma$ is the adiabatic constant for the gas, $P$ the gas pressure, and $\rho$ the gas density; other formulas are equivalent. Note that intensity and frequency do not appear in this equation. Hence sound is non-dispersive over wide ranges, given stable atmospheric conditions.
Sometimes thunder is given as a counter-example, where a variety of sounds are heard following a lightning strike but this is not due to dispersion; rather it is the multiple branches of the pre-strike, the main strike, and the extended distances covered by the lightning, plus, sometimes, echos.
Light is similarly non-dispersive in the ordinary atmosphere, but changes in pressure, temperature, and humidity change that, hence mirages.